Sometimes, stress builds within the earth to such a significant level that it must be relieved, typically by rupture or failure of material. This rupture, an earthquake, may release enormous amounts of energy in the form of heat and seismic waves and may result in significant displacement of land masses at the surface. During an earthquake, a very significant amount of energy is released in a period of time that may range from a fraction of a second to several seconds. This energy release can be compared to detonating a large explosive charge underground, as the effects of both share several similarities. Energy released quickly during such events, produces a pressure pulse which radiates from the point of origin as stress waves. The size, profile and depth of the energy release area has an important bearing on the frequency of the vibrations. In the case of underground explosive detonations, if the release area is larger, and/or deeper, the frequency will be lower. If the release area is smaller, and/or closer to the surface, the frequency will be higher.
A pressure wave may move sonic or supersonic through the material it transits. A shock wave refers only to a pressure wave which moves faster than the sound speed of the material through which it transits. In this document, "stress waves" refer to pressure waves transiting a material at the sonic velocity of that material, and "shock waves" refers to pressure waves transiting a material above the sonic velocity of that material. Pressure waves and shock waves are traveling pressure fluctuations which cause local compression of the material through which they transit. Stress waves cause disturbances whose gradients, or rates of displacement are small on the scale of the displacement itself. Stress waves travel at a speed determined by the characteristic of a given medium and therefore must be referred to a particular subject medium.
Shock waves are distinguished from stress waves in two key respects. First, shock waves travel faster than the sonic velocity of the medium through which they transit. Secondly, local displacements of atoms or molecules comprising a medium that is being transited by a shock wave are much larger than those produced by stress waves. Together, these two factors produce gradients or rates of displacement much larger than the local fluctuations themselves.
Energy is required to produce pressure waves and once the driving source ceases to produce the stress disturbances, the waves decay. Absorption and attenuation involve acceleration of the natural damping process, which therefore means removing energy from the pressure waves. All matter through which pressure waves travel, naturally attenuate these waves by virtue of their inherent mass. Materials possess different acoustic attenuating properties, strongly affected by density and the presence or absence of phase boundaries and structural discontinuities. For example, porous solid materials are better attenuators of stress waves than perfect crystalline solids.
Acoustic impedance is the product of a material's sonic velocity multiplied by the material's mass per unit area. A material's acoustic impedance indicates how well it will transmit pressure waves. The higher the value, the greater (higher amplitude and/or higher velocity) the stress transmission in that particular material. Water has a density of 1 gram/cc, while air has a density of 1/1000 that of water. Water has a sonic velocity of approximately 1650 meters per second and air has a sonic velocity of 344 meters per second. The ratio between the acoustic impedance of water to air is nearly 4,800. Different types of rock will have varying sonic velocities due to differences in densities, crystallographic structure and the presence of discontinuities.
During a large explosion in a solid (such as rock), and during an earthquake, the resultant pressure pulse is a series of waves. There are two main types of body waves originating from the interior of the solid, which have different particle motions and velocities. The first wave to arrive (i.e. fastest), at a given point from the origin of the energy release, is a compressional wave, usually called a "P-wave". The particle motion in the P-wave is a "push-pull" motion, radially away and toward the origin, or in other words parallel to the direction of wave propagation. The other wave is a shear wave, usually referred to as an "S-wave". S-waves are generally transversal waves and the article motion is perpendicular to the travel path. S-waves and other waves will arrive after the P-wave because they are slower. P-waves and S-waves are both volume waves since they propagate in a three-dimensional space. At interfaces between different media (for instance, at interfaces between ground and air, between ground and water or between layers of ground of very different elastic characteristics), different types of surface waves are developed.
During an earthquake, when the P-waves and S-waves arrive at the ground surface, other waves are also developed. The two primary surface waves are known as "Love waves" and "Rayleigh waves".
The first, and faster of the two, are Love waves, whose motion is essentially the same as that of S-waves without vertical displacement. Love waves move the ground from side to side in a horizontal plane parallel to the earth's surface but transverse to the direction of propagation. The second, most prominent and common surface waves, are Rayleigh waves, or "R-waves" (elastic wave). P-waves, S-waves and R-waves produce vertical motion, whereas Love waves produce only horizontal motion. Rayleigh waves, because of their vertical component of motion, can affect bodies of water such as lakes, whereas Love waves (which do not propagate through water) can affect surface water only at the sides of lakes, water reservoirs and ocean bays, by a movement backwards and forwards, pushing the water sideways like the sides of a vibrating tank. The Love surface waves are the third to arrive because they travel slower than P-waves and S-waves. In the Rayleigh wave, the particles are described in a retrograde elliptical motion. The vertical component of the particle motion as its maximum just below the surface, but thereafter diminishes relatively rapidly with depth. Rayleigh waves may be compared to waves generated when a rock is thrown into a pond.
Because the various waves travel at different velocities, the differences in their arrival times at the land mass depend on the distance traveled from their origin. Very near the origin, the waves are mixed and are indistinguishable from one another.
As the distance from the point of origin increases, the waves separate and it is possible to see the differences in their characteristics. If all three wave types are well developed, the P-wave has the highest frequency and the smallest particle motion; the S-wave has a lower frequency and larger particle motion; and the R-wave has a frequency still lower and a particle displacement that is still larger in amplitude.
Propagation of Elastic Waves
The elasticity of a homogeneous, isotropic solid can be identified by two constants, k and .mu..
k is the modulus of incompressibility or bulk modulus ##EQU1## for granite, k is about 27.times.10.sup.10 dynes per cm.sup.2 for water, k is about 2.0.times.10.sup.10 dynes per cm.sup.2
.mu. is the modulus of rigidity ##EQU2## for granite, .mu. is about 1.6.times.10.sup.11 dynes per cm.sup.2 for water, .mu. is 0
Within the body of an elastic solid with a density .rho., two elastic waves can propagate:
______________________________________ P-waves Velocity for granite, .alpha. = 5.5 km/sec for water, .alpha. = 1.5 km/sec S-waves Velocity for granite, .beta. = 3.0 km/sec for water, .beta. = 0.0 km/sec ______________________________________
Along the free surface of an elastic solid, two surface elastic waves can propagate:
Rayleigh waves Velocity C.sub.r &lt;0.92 .beta. where .beta. is the S wave velocity of rock.
Love waves (for layered solid) Velocity .beta..sub.1 &lt;C.sub.L &lt;.beta..sub.2 where .beta..sub.1 and .beta..sub.2 are S-wave velocities of the surface and deeper layers, respectively.
The dimensions of a harmonic wave are measured in terms of period T and wavelength .lambda.. EQU Wave velocity v=.lambda./T (3) EQU Wave frequency f=1/T (4)
As a particle vibrates, its motion can be described in several different ways. It moves a certain distance from its resting position, which is termed "displacement". It moves in a repetitive cycle or oscillation a certain number of times each second, which is termed frequency, and is usually expressed in Hertz. Displacement alone does not express the intensity of a motion.
Something can move a great distance very slowly and not be damaged by that displacement. To assess damage potential, the rate or velocity of the displacement must be taken into account.
For simple harmonic motion, the relationship between displacement, frequency and velocity allows calculation of any of the three if the other two are known. EQU V=2 .pi.f D (5)
where V=peak particle velocity in inches per second (ips) PA1 .pi.=3.14 PA1 f=frequency in Hertz (cycles per second) PA1 D=maximum displacement (inches) PA1 also T=period=1/f PA1 and 2 .pi.f=the circular frequency or angular velocity of the particle PA1 Hence, D=V/2 .pi.f and f=V/2 .pi.D PA1 h=distance between water surface and lowest level of the structure ##EQU8## where C.sub.y =is a parabolic function of depth of water a.sub.h =peak horizontal ground acceleration from quake PA1 w 32 specific gravity of water PA1 h=height ##EQU9##
As mentioned earlier, a shock wave is a pressure wave which is transiting a material at a speed greater than its sonic velocity. This wave produces an abrupt pressure "jump" in the material. U.sub.s (shock velocity)&gt;C.sub.B (bulk sound velocity), which means that U.sub.s is supersonic with respect to the material (in its initial state). Compressional shock waves act to accelerate the particles of a material in the direction of wave propagation. On the other hand, rarefraction waves (expansion, unloading waves) act to accelerate the particles of a material in the opposite direction of wave propagation. Rarefraction waves may also be known as reflection waves, as they are a result of a compression wave being reflected back towards its point of origin as a tensile wave.
In the case of shock waves, jump relations (which describe the changes across the shock front) are obtained from the conservation of mass, momentum and energy, known as equations of state or E.O.S. The Rankine-Hugoniot E.O.S. for shock waves are: EQU Mass: .rho..sub.0 U.sub.s =.rho..sub.1 (U.sub.s -up) (6) EQU Momentum: P.sub.1 -P.sub.0 .rho..sub.0 U.sub.s up (7) ##EQU3## When p.sub.1 &gt;&gt;p.sub.0, Equations (5) and (6) combine to give ##EQU4## During an earthquake structurally damaging energy may be transmitted through the ground at speeds higher than four kilometers per second. Ground motions from earthquakes are characterized by large displacements, low frequencies and long durations. Stress waves will be transmitted from the earth into a body of water and then traverse the body of water until they encounter another medium or material (which may be a structure).
When this stress wave hits a structure, it imparts particle velocity into the materials of the structure.
The term "coupling" describes the interface between two different (dissimilar) materials. The amount of coupling between materials is a function of area joining the different materials, the bond between the two materials, and a function of the respective a acoustic impedances of the two materials, as well as the direction of displacement of the stress waves.
As a spherical shock wave is transmitted into the medium departing from the point of origin (energy release) the shock wave amplitude and energy decrease with distance. For very high shock pressures, the deformation of the material accompanying the one-dimensional shock compression is plastic. But as the shock propagates radially from the point of origin, the amplitude decreases very quickly and soon reaches the limit termed the Hugoniot elastic limit (HEL). From then on, the deformation is purely elastic. Such elastic compression waves are stress wave, and they propagate at the sonic velocity of the material being transited.
When the stress wave travels into a new medium with a different acoustic impedance, part of the energy will be reflected and another part will be transmitted. Pressure within a structure will be called stress rather than pressure, and will be designated ".sigma.". To reiterate, the impedance of a medium is given by the product of the density .rho. and the sonic velocity c. Consider an elastic infinite medium through which a plane stress wave passes. The stress induced .sigma..sub.1 is the product of the density .rho..sub.1, the sound velocity c.sub.1, and the particle velocity v.sub.1, EQU .sigma..sub.1 =.rho..sub.1 c.sub.1 v.sub.1 (9)
which clearly stems from the conservation of momentum M (equation 7) in the Rankine-Hugoniot E.O.S. In general, if a plane compressive stress wave reaches a boundary which is not parallel to the wave front, four waves are generated. Two of these are reflected waves, moving back into the medium from which the original wave came, a shear wave and a compression or expansion wave; the other two waves, also a shear wave and an expansion or compression wave, are transmitted into the new medium.
Consider a simpler, special case when the stress wave has a normal incidence to the boundary. Then, a wave with stress level .sigma..sub.R and particle velocity v.sub.R is reflected back. Another wave is transmitted into the second medium which is assumed to have density .rho..sub.2. It has stress level .sigma..sub.T, particle velocity v.sub.T, and shock wave velocity c.sub.2.
According to equation (9), it follows that: ##EQU5##
The following conditions must be fulfilled assuming that the two materials are in contact with each other during the shock wave passage: EQU .sigma..sub.1 +.sigma..sub.R =.sigma..sub.T EQU v.sub.1 +v.sub.R =v.sub.T (11)
Combining equations (10) and (11) gives the following expressions for the stress levels of the reflected and the transmitted waves: ##EQU6##
From equations (12) and (13), it is apparent that the ratio .mu. between the impedances varies. If the stress wave travels toward a medium with the same impedance (.mu.=1), no reflection occurs (.sigma..sub.R /.sigma..sub.1 =0).
When a stress wave passes from rock to air, or more specifically, from rock or water to air, gas or foam, (.rho..sub.1 c.sub.1 &gt;&gt;.rho..sub.2 c.sub.2), i.e., .mu. is very large, so almost no energy is transmitted. If .rho..sub.1 c.sub.1 &gt;.rho..sub.2 c.sub.2, i.e., .mu.&gt;1, then the reflected compression wave will appear as a tensile wave. Finally, if .mu.&lt;1, then the reflected wave is a compression wave.
The relationship between vibrations and damage to a structure is complicated for many reasons. Some structures are more solidly built than others, and have different dimensions, materials, methods of assembly, and types of foundations. Moreover, the intensity, type, frequency range and wavelength of the vibrations, and the direction of incidence of the wave fronts relative to the main axis of the structure all play important parts in the origin of damage.
In concrete structures, such as a dam, two causes account for added stresses during an event like an earthquake: the acceleration of the mass of the structure and the changes of water pressures.
There are two distinct water pressures which affect a structure in contact with water and which actg simultaneously during an earthquake. The first is hydrostatic pressure (due to the depth) P.sub.1, which is present before, during and after the earthquake. The second is hydrodynamic pressure (due to ground acceleration transferring energy into the water and the stress waves transiting the water and interacting with the structure) P.sub.2, which is caused by the earthquake and is not normally present. ##EQU7## where .zeta.=water density=1000 kg/m.sup.3 g=gravity acceleration.congruent. 10 M/s.sup.2
Hydrodynamic forces may be absorbed and attenuated very effectively through adiabatic compression of gas bubbles. As the pressure increases within the gas it will heat. The heat causes the gas to expand. If the pressure is still higher outside the bubbles, interface, it will be compressed again and then expand.
In the past various attempts have been made to protect structures from the effects of shock waves and stress waves from earthquakes, explosions and other large energy sources.
U.S. Pat. No. 5,174,082 shows material described as an "island" with mechanical properties different than that of the ground. The islands are anchored deep underground by cable. A variant listed is to inter-disperse wells 5 m to 30 m deep filled with a granular or pulverized material, among the islands.
U.S. Pat. No. 5,173,012 shows a vertical wall barrier between a rail line and a building. The barrier is intended to stop ground-borne noise and vibration from travelling through the ground. It is constructed of two parallel concrete walls with elastic mat sandwiched between the walls.
U.S. Pat. No. 4,484,423 teaches a trench intended to be as deep as possible (but at least 100 meters deep), installed near a ground based structure to be protected (perhaps 3-60 meters in the case of a conventional power station). The preferred fill in the trench is a liquid or other material with a low shear, or gas bags or other media which does not allow S-waves. This technique is obviously impractical for many reasons, especially in submarine applications involving a dam.
None of these prior techniques can protect a structure in contact with water or a submerged structure, from energy being transferred to it through the water. Further, these methods do not lend themselves to protecting the submarine portions of pre-existing structures from pressure waves.
Canadian Patent No. 2,699,117 asserts that in the context of submarine blasting, interposing an air curtain of reasonable density between the structure to be protected and the source of waves, the resulting pressures can be reduced by 90%.
U.S. Pat. No. 5,394,786 teaches the use of aqueous foam as a buffer medium to attenuate S-waves in the ground. Aqueous foam might be useful when attempting to attenuate S-waves in the ground but is of no use in submarine applications. No attenuation will be present in such applications because the impedance of the aqueous foam will be nearly identical to that of the water.